A327354 - OEIS

%I #5 Sep 11 2019 20:22:02

%S 1,1,2,8,53,747,45156,54804920,19317457655317

%N Number of disconnected or empty antichains of nonempty subsets of {1..n} (non-spanning edge-connectivity 0).

%C An antichain is a set of sets, none of which is a subset of any other.

%C The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

%F Equals the binomial transform of the exponential transform of A048143 minus A048143.

%e The a(1) = 1 through a(3) = 8 antichains:

%e   {}  {}         {}

%e       {{1},{2}}  {{1},{2}}

%e                  {{1},{3}}

%e                  {{2},{3}}

%e                  {{1},{2,3}}

%e                  {{2},{1,3}}

%e                  {{3},{1,2}}

%e                  {{1},{2},{3}}

%t csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];

%t stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];

%t Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Length[csm[#]]!=1&]],{n,0,4}]

%Y Column k = 0 of A327353.

%Y The covering case is A120338.

%Y The unlabeled version is A327426.

%Y The spanning edge-connectivity version is A327352.

%Y Cf. A014466, A326787, A327071, A327148, A327236, A327355, A327357.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 10 2019